To find the product of \(\sqrt{12}\) and \(7\sqrt{10}\), we can multiply them directly.
First, let's express the product:
\[ \sqrt{12} \cdot 7\sqrt{10} \]
This can be simplified as:
\[ 7 \cdot \sqrt{12} \cdot \sqrt{10} \]
Next, we can combine the square roots:
\[ 7 \cdot \sqrt{12 \cdot 10} \]
Now, we calculate \(12 \cdot 10\):
\[ 12 \cdot 10 = 120 \]
So, we have:
\[ 7 \cdot \sqrt{120} \]
Next, we simplify \(\sqrt{120}\). We factor \(120\) to find perfect squares:
\[ 120 = 4 \cdot 30 = 4 \cdot (2 \cdot 3 \cdot 5) \]
Since \(4\) is a perfect square, we can take its square root out of the radical:
\[ \sqrt{120} = \sqrt{4 \cdot 30} = \sqrt{4} \cdot \sqrt{30} = 2\sqrt{30} \]
Now substituting back, we get:
\[ 7 \cdot \sqrt{120} = 7 \cdot 2\sqrt{30} = 14\sqrt{30} \]
Thus, the product \(\sqrt{12} \cdot 7\sqrt{10}\) in simplest radical form is:
\[ \boxed{14\sqrt{30}} \]
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